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Revista Colombiana de Estadística Junio 2014, volumen 37, no. 1, pp. 223 a 243 The Poisson-Lomax Distribution Distribución Poisson-Lomax Bander Al-Zahrani a , Hanaa Sagor b Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia Abstract In this paper we propose a new three-parameter lifetime distribution with upside-down bathtub shaped failure rate. The distribution is a com- pound distribution of the zero-truncated Poisson and the Lomax distribu- tions (PLD). The density function, shape of the hazard rate function, a general expansion for moments, the density of the rth order statistic, and the mean and median deviations of the PLD are derived and studied in de- tail. The maximum likelihood estimators of the unknown parameters are obtained. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance-covariance matrix. Finally, a real data set is analyzed to show the potential of the new proposed distribution. Key words : Asymptotic variance-covariance matrix, Compounding, Life- time distributions, Lomax distribution, Poisson distribution, Maximum like- lihood estimation. Resumen En este artículo se propone una nueva distribución de sobrevida de tres parámetros con tasa fallo en forma de bañera. La distribución es una mezcla de la Poisson truncada y la distribución Lomax. La función de densidad, la función de riesgo, una expansión general de los momentos, la densidad del r-ésimo estadístico de orden, y la media así como su desviación estándar son derivadas y estudiadas en detalle. Los estimadores de máximo verosímiles de los parámetros desconocidos son obtenidos. Los intervalos de confianza asintóticas se obtienen según la matriz de varianzas y covarianzas asintótica. Finalmente, un conjunto de datos reales es analizado para construir el po- tencial de la nueva distribución propuesta. Palabras clave : mezclas, distribuciones de sobrevida, distribució n Lomax, distribución Poisson, estomación máximo-verosímil. a Professor. E-mail: [email protected] b Ph.D student. E-mail: [email protected] 223

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Page 1: The Poisson-Lomax Distribution€¦ · The Poisson-Lomax Distribution Distribución Poisson-Lomax Bander Al-Zahrania, Hanaa Sagorb Department of Statistics, King Abdulaziz University,

Revista Colombiana de EstadísticaJunio 2014, volumen 37, no. 1, pp. 223 a 243

The Poisson-Lomax Distribution

Distribución Poisson-Lomax

Bander Al-Zahrania, Hanaa Sagorb

Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia

Abstract

In this paper we propose a new three-parameter lifetime distributionwith upside-down bathtub shaped failure rate. The distribution is a com-pound distribution of the zero-truncated Poisson and the Lomax distribu-tions (PLD). The density function, shape of the hazard rate function, ageneral expansion for moments, the density of the rth order statistic, andthe mean and median deviations of the PLD are derived and studied in de-tail. The maximum likelihood estimators of the unknown parameters areobtained. The asymptotic confidence intervals for the parameters are alsoobtained based on asymptotic variance-covariance matrix. Finally, a realdata set is analyzed to show the potential of the new proposed distribution.

Key words: Asymptotic variance-covariance matrix, Compounding, Life-time distributions, Lomax distribution, Poisson distribution, Maximum like-lihood estimation.

Resumen

En este artículo se propone una nueva distribución de sobrevida de tresparámetros con tasa fallo en forma de bañera. La distribución es una mezclade la Poisson truncada y la distribución Lomax. La función de densidad, lafunción de riesgo, una expansión general de los momentos, la densidad delr-ésimo estadístico de orden, y la media así como su desviación estándar sonderivadas y estudiadas en detalle. Los estimadores de máximo verosímilesde los parámetros desconocidos son obtenidos. Los intervalos de confianzaasintóticas se obtienen según la matriz de varianzas y covarianzas asintótica.Finalmente, un conjunto de datos reales es analizado para construir el po-tencial de la nueva distribución propuesta.

Palabras clave: mezclas, distribuciones de sobrevida, distribució n Lomax,distribución Poisson, estomación máximo-verosímil.

aProfessor. E-mail: [email protected] student. E-mail: [email protected]

223

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224 Bander Al-Zahrani & Hanaa Sagor

1. Introduction

Marshall & Olkin (1997) introduced an effective technique to add a new pa-rameter to a family of distributions. A great deal of papers have appeared inthe literature used this technique to propose new distributions. In their paper,Marshall & Olkin (1997) generalized the exponential and Weibull distributions.Alice & Jose (2003) followed the same approach and introduced Marshall-Olkinextended semi-Pareto model and studied its geometric extreme stability. Ghitany,Al-Hussaini & Al-Jarallah (2005) studied the Marshall-Olkin Weibull distributionand established its properties in the presence of censored data. Marshall-Olkinextended Lomax distribution was introduced by Ghitany, Al-Awadhi & Alkhalfan(2007). Compounding Poisson and exponential distributions have been consideredby many authors; e.g. Kus (2007) proposed the Poisson-exponential lifetime distri-bution with a decreasing failure rate function. Al-Awadhi & Ghitany (2001) usedthe Lomax distribution as a mixing distribution for the Poisson parameter and ob-tained the discrete Poisson-Lomax distribution. Cancho, Louzada-Neto & Barriga(2011) introduced another modification of the Poisson-exponential distribution.

Let Y1, Y2, . . . , YZ be independent and identically distributed random variableseach has a density function f , and let Z be a discrete random variable having azero-truncated Poisson distribution with probability mass function

PZ(z) ≡ PZ(z, λ) =e−λλz

z!(1− e−λ), z ∈ {1, 2, . . .}, λ > 0. (1)

Suppose that X is a random variable representing the lifetime of a parallel-systemof Z components, i.e. X = max{Y1, Y2, . . . , Yz}, and Y ’s and Z are independent.The conditional distribution function of X|Z has the probability density function(pdf)

fX|Z(x|z) = zf(x)[F (x)]z−1. (2)

where F (x) is the cumulative distribution function (cdf) corresponding to f(x).

A compound probability function (pdf) of fX|Z(x|z) and PZ(z), where X is acontinuous random variable (r.v.) and Z a discrete r.v. is defined by

gX(x) =

∞∑z=1

fX|Z(x|z)PZ(z). (3)

Substitution of (1) and (2) in (3) then yields

gX(x) =

∞∑z=1

zf(x)[F (x)]z−1

(λze−λ

z!(1− e−λ)

)=

λf(x)e−λ(1−F (x))

(1− e−λ), x > 0, λ > 0.

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The Poisson-Lomax Distribution 225

The reliability and the hazard rate functions of X are, respectively, given by

G(x, λ) =1− e−λF (x)

(1− e−λ), x > 0, (4)

hG(x, λ) =λf(x)e−λF (x)

1− e−λF (x)=

λf(x)

eλF (x) − 1. (5)

In this paper we propose a new lifetime distribution by compounding Poissonand Lomax distributions. As we have mentioned in the previous chapters, theLomax distribution with two parameters is a special case of the generalized Paretodistribution, and ti is also known as the Pareto of the second type. A randomvariable X is said to have the Lomax distribution, abbreviated as X ∼ LD(α, β),if it has the pdf

fLD(x;α, β) = αβ (1 + βx)−(α+1)

, x > 0, α, β > 0. (6)

Here α and β are the shape and scale parameters, respectively. Analogous tuabove, the survival and hazard functions associated with (6) are given by

FLD(x;α, β) = (1 + βx)−α

, x > 0, (7)

hLD(x;α, β) =αβ

1 + βx, x > 0. (8)

The rest of the paper is organized as follows. In Section 2, we give explicit formsand interpretation for the distribution function and the probability density func-tion. In Section 3, we discuss the distributional properties of the proposed dis-tribution. Section 4 discusses the estimation problem using the maximum likeli-hood estimation method. In Section 5, an illustrative example, model selections,goodness-of-fit tests for the distribution with estimated parameters are all pre-sented. Finally, we conclude in Section 6.

2. Model Formulation

Substitution of (7) in (4) yields the following reliability function:

G(x;α, β, λ) =1− e−λ(1+βx)−α

(1− e−λ), x > 0, α, β, λ > 0. (9)

The pdf associated with (9) is expressed in a closed form and is given by

g(x;α, β, λ) =αβλ (1 + βx)

−(α+1)e−λ(1+βx)−α

(1− e−λ), x > 0, α, β, λ > 0. (10)

The density function given by (10) can be interpreted as a compound of the zero-truncated Poisson distribution and the Lomax distribution. Suppose that X =max{Y1, Y2, · · · , Yz}, and each Y is distributed according to the Lomax distribtion.

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226 Bander Al-Zahrani & Hanaa Sagor

The variable Z has zero-truncated Poisson distribution and the variables Y ’s andZ are independent. Then the conditional distribution function of X|Z has the pdf

fX|Z(x|z;α, β) = zαβ(1 + βx)−(α+1)[1− (1 + βx)−α]z−1. (11)

The joint distribution of the random variables X and Z, denoted by fX,Z(x, z), isgiven by

fX,Z(x, z) =z

z!(1− e−λ)αβ(1 + βx)−(α+1)[1− (1 + βx)−α]z−1e−λλz, (12)

the marginal pdf of X is as follows.

fX(x;α, β, λ) =αβλe−λ(1 + βx)−(α+1)

(1− e−λ)

∞∑z=1

[(1− (1 + βx)−α)λ]z−1

(z − 1)!

=αβλe−λ(1 + βx)−(α+1)eλ(1−(1+βx)−α)

(1− e−λ)

=αβλ(1 + βx)−(α+1)e−λ(1+βx)−α

(1− e−λ),

which is the distribution with the pdf given by (10). The distribution of X may bereferred to as the Poisson-Lomax distribution. Symbolically it is abbreviated byX ∼ PLD(α, β, λ) to indicate that the random variable X has the Poisson-Lomaxdistribution with parameters α, β and λ.

3. Distributional Properties

In this section, we study the distributional properties of the PLD. In particular,if X ∼ PLD(α, β, λ) then the shapes of the density function, the shapes of thehazard function, moments, the density of the rth order statistics, and the meanand median deviations of the PLD are derived and studied in detail.

3.1. Shapes of pdf

The limit of the Poisson-Lomax density as x→∞ is 0 and the limit as x→ 0is αβλ/(eλ − 1). The following theorem gives simple conditions under which thepdf is decreasing or unimodal.

Theorem 1. The pdf, g(x), of X ∼ PLD(α, β, λ) is decreasing (unimodal) if thefunction ξ(x) ≥ 0 (< 0) where ξ(x) = α(1− λ(1 + βx)−α) + 1, independent of β.

Proof . The first derivative of g(x) is given by

g′(x) = − αβ2λ

1− e−λ(1 + βx)−(α+2) e−λ(1+βx)−αξ((1 + βx)−α),

where ξ(y) = α(1−λy) + 1, and y = (1 +βx)−α < 1. Then we have the following:

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The Poisson-Lomax Distribution 227

(i) If ξ(1) = α(1− λ) + 1 > 0, then ξ(y) > 0 for all y < 1, and hence, g′(x) ≤ 0for all x > 0, i.e. the function g(x) is decreasing.

(ii) If ξ(1) < 0, then ξ(y) has a unique zero at yξ = α+1αλ < 1 . Since y =

(1 + βx)−α is one to one transformation, it follows that g(x) has also aunique critical point at xg = 1

β (y−1/αξ − 1).

Finally, since g(0) = αβλ/(eλ − 1) and g(∞) = 0 then xg must be a point ofabsolute maximum for g(x).

Note 1. It should be noted that:

(i) When λ ∈ (0, 1], g(x) is decreasing in x > 0 for all values of α, β > 0.

(ii) When λ > 1, g(x) may still exhibit a decreasing behavior, depending on thevalues of α, λ such that α(1− λ) + 1 > 0.

(iii) The mode of the Poisson-Lomax distribution is given by

Mode(x) =

0, if α(1− λ) + 1 ≥ 0,

[(αλα+1

)1/α

− 1

]otherwsie.

(13)

Figure 1 shows the pdf curves for the PLD(α, β, λ) for selected values of theparameters α, β and λ. From the curves, it is quite evident that the PLD ispositively skewed distribution. It becomes highly positively skewed for large valuesof the involved parameters.

3.2. Hazard Rate Function

The hazard rate function (hrf) of a random variable X is defined by h(x) =f(x)/F (x), where F = 1− F . The hazard function of X ∼ PLD(α, β, λ) is givenby

h(x) =αβλ(1 + βx)−(α+1)

eλ(1+βx)−α − 1, x > 0. (14)

The following theorem gives simple conditions under which the hrf, given in (14),is decreasing or unimodal.

Theorem 2. The hrf, h(x), of X ∼ PLD(α, β, λ) is decreasing (unimodal) ifη(x) ≥ 0(< 0) where η(x) = −(α + 1) + (α + 1 − αλ(1 + βx)−α) eλ(1+βx)−α ,independent of β.

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228 Bander Al-Zahrani & Hanaa Sagor

0 2 4 6 8 100.

00.

20.

40.

6

β = 0.5,λ =0.5

x

Pro

babi

lity

Den

sity

Fun

ctio

n

α =0.5α =1α =2

0 2 4 6 8 10

0.0

0.2

0.4

0.6

α =0.5,λ =0.5

x

Pro

babi

lity

Den

sity

Fun

ctio

n

β = 0.5β = 1β = 1.5

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

α =2,β = 0.5

x

Pro

babi

lity

Den

sity

Fun

ctio

n

λ = 2λ = 4λ = 8

0 2 4 6 8 100.

00.

51.

01.

5

β = 1,λ =6

x

Pro

babi

lity

Den

sity

Fun

ctio

n

α =2α =4α =6

Figure 1: Plot of the probability density function for different values of the parametersα, β and λ.

Proof . The first derivative of h(x) with respect to x is given by

h′(x) =−αβ2λ(1 + βx)−(α+2)

(eλ(1+βx)−α − 1)2

[(α+ 1) (eλ(1+βx)−α − 1)− αλ(1 + βx)−αeλ(1+βx)−α

]=−αβ2λ(1 + βx)−(α+2)

(eλ(1+βx)−α − 1)2

[(α+ 1− αλ(1 + βx)−α) eλ(1+βx)−α − (α+ 1)

]=−αβ2λ(1 + βx)−(α+2)

(eλ(1+βx)−α − 1)2η((1 + βx)−α),

where η(y) = −(α+1)+(α+1−αλy)eλy, and y = (1+βx)−α < 1. The remainingof the proof is similar to that of Theorem 1.

Note 2. The following should be noted.

(i) For λ ∈ (0, 1], h(x) is decreasing in x > 0 for all values of α, β > 0.

(ii) For λ > 1, h(x) may still exhibit a decreasing behavior, depending on thevalues of α and λ such that (1 + (1− λ)α)eλ − (α+ 1) ≥ 0.

(iii) Since (1 + (1 − λ)α)eλ − (α + 1) ≥ 0 implies that α(1 − λ) + 1 ≥ 0, then adecreasing hrf implies decreasing pdf. The converse is not necessarily true,e.g. α = 2, λ = 2 implies decreasing pdf but unimodal hrf.

Revista Colombiana de Estadística 37 (2014) 223–243

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The Poisson-Lomax Distribution 229

(iv) Since (1 + (1 − λ)α)eλ − (α + 1) < 0 implies that α(1 − λ) + 1 < 0, thena unimodal pdf implies unimodal hrf. The converse is not necessarily true,e.g., α = 2, λ = 2 implies unimodal hrf but decreasing pdf.

Figure 2 shows the hrf curves for the PLD(α, β, λ) for selected values of theparameters α, β and λ.

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

β = 0.5,λ =0.5

x

Haz

ard

Func

tion

α =0.5α =1α =2

0 2 4 6 8 10

0.0

0.2

0.4

0.6

α =0.5,λ =0.5

x

Haz

ard

Func

tion

β = 0.5β = 1β = 1.5

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

α =2,β = 0.5

x

Haz

ard

Func

tion

λ =2λ =4λ =8

0 2 4 6 8 10

0.0

1.0

2.0

3.0

β = 1,λ =6

x

Haz

ard

Func

tion

α =2α =4α =6

Figure 2: Plot of the hazard function for different values of the parameters α, β and λ.

3.3. Moments

We present an infinite sum representation for the rth moment, µ′r = E [Xr],and consequently the first four moments and variance for the PLD.

Theorem 3. The rth moment about the origin of a random variable X, whereX ∼ PLD(α, β, λ), and α, β, λ > 0, is given by the following:

µ′r = E [Xr] =α

βr(1− e−λ)

∞∑n=0

r∑j=0

(r

j

)λn+1(−1)n+r−j+1

(j − α(n+ 1))n!, r = 1, 2, . . . (15)

Proof . The rth moment of X can be determined by direct integration using thepdf, i.e. µ′r =

∫xrf(x)dx. We use the Maclaurin expansion of ex =

∑∞n=0 x

n/n!, for all x.We also use the series representation

(1− w)k =

k∑j=0

(k

j

)(−1)j wj , where k is a positive integer.

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230 Bander Al-Zahrani & Hanaa Sagor

Therefore, after some transformations and integrations we have

E [Xr] =

∫ ∞0

xrαβλ (1 + βx)

−(α+1)e−λ(1+βx)−α

(1− e−λ)dx.

Setting y = 1 + βx, dx = dy/β yields

E [Xr] =αλ

βr(1− e−λ)

∫ ∞1

(y − 1)ry−(α+1) e−λy−αdy

=αλ

βr(1− e−λ)

∫ ∞1

r∑j=0

(r

j

)yj−α−1(−1)r−j

∞∑n=0

(−λy−α)n

n!

dy

=αλ

βr(1− e−λ)

∫ ∞1

∞∑n=0

r∑j=0

(r

j

)λn+1(−1)n+r−jyj−α(n+1)−1

n!dy

βr(1− e−λ)

∞∑n=0

r∑j=0

(r

j

)λn+1(−1)n+r−j+1

(j − α(n+ 1))n!.

This completes the proof of the theorem.

An alternative representation formula for (15) can readily be found by expand-ing and substituting in the binomial expansion.

µ′r =r!

βr(1− e−λ)

∞∑k=1

(−1)k+r−1λk

k!(1− kα) · · · (r − kα), α 6= i

k, i = 1, 2, · · · (16)

One may use this representation to obtain the mean and the variance of X.

Corollary 1. Let X ∼ PLD(α, β, λ), where α, β, λ > 0. Then the first fourmoments of X are given, respectively, as follows:

µ = E [X] = 1β(1−e−λ)

∑∞k=1

(−1)kλk

k!(1−kα) ,

µ′2 = E [X2] = 2β2(1−e−λ)

∑∞k=1

(−1)k+1λk

k!(1−kα)(2−kα) ,

µ′3 = E [X3] = 6β3(1−e−λ)

∑∞k=1

(−1)k+2λk

k!(1−kα)(2−kα)(3−kα) ,

µ′4 = E [X4] = 24β4(1−e−λ)

∑∞k=1

(−1)k+3λk

k!(1−kα)(2−kα)(3−kα)(4−kα) .

(17)

Proof . Applying relations (15) or (16) for r = 1, 2, 3 and r = 4 yields the desiredresults.

Based on the results given in relations (17), the variance of X, denoted byσ2 = µ′2 − µ2 is given by

σ2 =2

β2(1− e−λ)

∞∑k=1

(−1)k+1λk

k!(1− kα)(2− kα)−

[1

β(1− e−λ)

∞∑k=1

(−1)kλk

k!(1− kα)

]2

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The Poisson-Lomax Distribution 231

It can be noticed from Table 1 that both the mean and the variance of the PLdistribution are decreasing functions of α and β but they are increasing in λ. Table2 shows the skewness and kurtosis of the PLD for various selected values of theparameters α, β and λ. The skewness is free of parameter β. Both the skewnessand kurtosis are decreasing functions of α and both are increasing of λ.

Table 1: Mean and variance of PLD for various values of α, β and λ.β = 0.5 β = 1.0 β = 2.0

λ α µ σ2 µ σ2 µ σ2

0.5 4.0 0.1184 1.6233 0.0592 0.4058 0.0296 0.10144.5 0.1013 1.1089 0.0506 0.2772 0.0253 0.06935.0 0.0885 0.8062 0.0442 0.2015 0.0221 0.05035.5 0.0785 0.6128 0.0392 0.1532 0.0196 0.03836.0 0.0706 0.4816 0.0353 0.1204 0.0176 0.0301

1.5 4.0 0.5890 1.9955 0.2945 0.4988 0.1472 0.12474.5 0.5018 1.3402 0.2509 0.3350 0.1254 0.08375.0 0.4369 0.9618 0.2184 0.2404 0.1092 0.06015.5 0.3869 0.7237 0.1934 0.1809 0.0967 0.04526.0 0.3471 0.5641 0.1735 0.1410 0.0867 0.0352

2.0 4.0 0.8104 2.0752 0.4052 0.5188 0.2026 0.12974.5 0.6892 1.377 0.3446 0.3442 0.1723 0.08605.0 0.5993 0.9791 0.2996 0.2447 0.1498 0.06115.5 0.5301 0.7313 0.2650 0.1828 0.1325 0.04576.0 0.4752 0.5668 0.2376 0.1417 0.1188 0.0354

4.0 4 1.4409 2.3195 0.7204 0.5798 0.3602 0.14494.5 1.2179 1.4705 0.6089 0.3676 0.3044 0.09195 1.0542 1.0089 0.5271 0.2522 0.2635 0.06305.5 0.9289 0.7322 0.4644 0.1830 0.2322 0.04576 0.8301 0.5542 0.4150 0.1385 0.2075 0.0346

3.4. L-moments

Suppose that a random sample X1, X2, . . . , Xn is collected from X ∼ PLD(θ),where θ = (α, β, λ). In what follows, we derive a general representation for theL-moments of X.

The rth population L-moments is given by

E[Xr:n] =

∫ ∞0

xf(Xr:n) dx

=

∫ ∞0

x

r−1∑i=0

n−r+i∑j=0

(r − 1

i

)(n− r + i

j

)(−1)i+j

{n!αβλ

(r − 1)!(n− r)!

× (1 + βx)−(α+1)e−λ(1+βx)−α(j+1)

(1− e−λ)n−r+i+1

}dx.

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232 Bander Al-Zahrani & Hanaa Sagor

Table 2: Skewness and kurtosis of PLD for various values of α, β and λ.β = 0.5 β = 1.0 β = 2.0

λ α γ3 γ4 γ3 γ4 γ3 γ40.5 4.5 3.6525 65.367 3.6525 16.3418 3.6525 4.0854

5.0 3.1739 24.114 3.1739 6.0285 3.1739 1.50715.5 2.8845 12.696 2.8845 3.1741 2.8845 0.79356.0 2.6904 7.8535 2.6904 1.9633 2.6904 0.49086.5 2.5510 5.3396 2.5510 1.3349 2.5510 0.3337

1.5 4.5 3.0490 75.423 3.049 18.855 3.0490 4.71395.0 2.5371 26.405 2.5371 6.6014 2.5371 1.65035.5 2.2239 13.345 2.2239 3.3362 2.2239 0.83406.0 2.0116 7.9879 2.0116 1.9969 2.0116 0.49926.5 1.8579 5.2867 1.8579 1.3216 1.8579 0.3304

2.0 4.5 3.0915 84.916 3.0915 21.229 3.0915 5.30725.0 2.5372 29.211 2.5372 7.3029 2.5372 1.82575.5 2.1963 14.561 2.1963 3.6404 2.1963 0.91016.0 1.9641 8.6212 1.9641 2.1553 1.9641 0.53886.5 1.7952 5.6554 1.7952 1.4138 1.7952 0.3534

4.0 4.5 3.8191 128.068 3.8191 32.017 3.8191 8.00425.0 3.1425 42.525 3.1425 10.631 3.1425 2.65785.5 2.7233 20.595 2.7233 5.1489 2.7233 1.28726.0 2.4357 11.905 2.4357 2.9764 2.4357 0.74416.5 2.2251 7.6554 2.2251 1.9138 2.2251 0.4784

Let y = (1 + βx) so x = (y − 1)/β and dx = (1/β)dy. After some transformation,we arrive to the formula:

E [Xr:n] =1

β

∞∑m=0

r−1∑i=0

n−r+i∑j=0

(j + 1)m(−λ)m+1 Aij(m+ 1)!(1− α(m+ 1))

, (18)

where Aij is

Aij =n!(−1)i+j

(r − 1)!(n− r)!(1− e−λ)n−r+i+1

(r − 1

i

)(n− r + i

j

).

One readily can use the relation (18) to obtain the first L-moments of Xr:n. Forexample, we take r = n = 1 to obtain λ1 = E [X1:1] which is the mean of therandom variable X.

λ1 = E[X1:1] =1

β(1− e−λ)

∞∑m=0

(−λ)m+1

(m+ 1)!(1− α(m+ 1)),

This result is consistent with that obtained in relation (17). The other two L-moments, λ2 and λ3, are respectively given by

λ2 = 1β

[∑∞m=0

∑1i=0

∑ij=0

(1i

)(ij

) (j+1)m(−1)i+j+m+1λm+1

(m+1)!(1−α(m+1))(1−e−λ)i+1

−∑∞m=0

∑1j=0

(1j

) (j+1)m(−1)j+m+1λm+1

(m+1)!(1−α(m+1))(1−e−λ)2

]

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The Poisson-Lomax Distribution 233

and

λ3 = 1β

[∑∞m=0

∑2i=0

∑ij=0

(2i

)(ij

) (j+1)m(−1)i+j+m+1λm+1

(m+1)!(1−α(m+1))(1−e−λ)i+1

−2∑∞m=0

∑1i=0

∑i+1j=0

(1i

)(i+1j

) 2(j+1)m(−1)i+j+m+1λm+1

(m+1)!(1−α(m+1))(1−e−λ)i+2

+∑∞m=0

∑2j=0

(2j

) 2(j+1)m(−1)j+m+1λm+1

(m+1)!(1−α(m+1))(1−e−λ)3

]

The method of L-moments consists of equating the first L-moments of a pop-ulation, λ1, λ2 and λ3, to the corresponding L-moments of a sample, l1, l2 and l3,thus getting a number of equations that are needed to be solved, numerically, interms of the unknown parameters, θ.

3.5. Order Statistics

Let X1, X2, . . . , Xn be a random sample of size n from the PL distributionin (10) and let X1:n, . . . , Xn:n denote the corresponding order statistics. Then,the pdf of Xr:n, 1 ≤ r ≤ n, is given by (see, David & Nagaraja 2003, Arnold,Balakrishnan & Nagaraja 1992)

g(r)(x) = Cr,ng(x)[G(x)]r−1[1−G(x)]n−r, 0 < x <∞, (19)

where Cr,n = [B(r, n− r + 1)]−1, with B(a, b) being the complete beta function.

Theorem 4. Let G(x) and g(x) be the cdf and pdf of a Poisson-Lomax distributionfor a random variable X. The density of the rth order statistic, say g(r)(x) is givenby

g(r)(x) = αβλCr,n

r−1∑i=0

n−r+i∑j=0

(r − 1

i

)(n− r + i

j

)(−1)i+j(1 + βx)−(α+1) e−λ(1+βx)−α(j+1)

(1− e−λ)n−r+i+1(20)

Proof . First it should be noted that (19) can be written as follows:

g(r)(x) = Cr,n

r−1∑i=0

(r − 1

i

)(−1)ig(x)[G(x)]n−r+i (21)

then the proof follows by replacing the reliability, G(x), and the pdf, g(x), of X ∼PLD(α, β, λ) which are obtained from (9) and (10), respectively, and substitutingthem into relation (21), and expanding the term (1− e−λ(1+βx)−α)n−r+i using thebinomial expansion.

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234 Bander Al-Zahrani & Hanaa Sagor

3.6. Quantile Function

Let X denote a random variable with the probability density function given by(10). The quantile function, denoted by Q(u), is

Q(u) = inf{x ∈ R : F (x) ≥ u}, where 0 < u < 1

By inverting the distribution function, F = 1− F , we can write the following:

Q(u) =1

β

[(− ln(u(1− e−λ) + e−λ)

λ

)−1/α

− 1

](22)

The first quartile, the median and the third quartile can be obtained simply byapplying (22). The quartiles; Q1 first quartile, Q2 second quartile, or the median,and Q3 third quartile are obtained in Table 3.

Table 3: The quartile values of the PLD for different values of α, β and λ.β = 0.5 β = 1.0 β = 2.0

λ α Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3

0.5 4.0 0.1870 0.4583 0.9647 0.0935 0.2291 0.4824 0.0467 0.1146 0.24124.5 0.1654 0.4025 0.8379 0.0827 0.2013 0.4189 0.0413 0.1006 0.20955.0 0.1482 0.3589 0.7403 0.0741 0.1794 0.3701 0.0371 0.0897 0.18515.5 0.1343 0.3238 0.6629 0.0672 0.1619 0.3315 0.0336 0.0809 0.16576.0 0.1228 0.2949 0.6002 0.0614 0.1474 0.3001 0.0307 0.0737 0.1500

1.5 4.0 0.2893 0.6431 1.2469 0.1446 0.3216 0.6234 0.0723 0.1608 0.31174.5 0.2552 0.5625 1.0767 0.1276 0.2813 0.5384 0.0638 0.1406 0.26925.0 0.2282 0.4998 0.9470 0.1141 0.2499 0.4735 0.0571 0.1249 0.23685.5 0.2065 0.4496 0.8450 0.1032 0.2248 0.4225 0.0516 0.1124 0.21126.0 0.1885 0.4086 0.7626 0.0942 0.2043 0.3813 0.0471 0.1021 0.1907

2.0 4.0 0.3521 0.7418 1.3856 0.1760 0.3709 0.6928 0.0880 0.1855 0.34644.5 0.3101 0.6474 1.1933 0.1550 0.3237 0.5966 0.0775 0.1618 0.29835.0 0.2770 0.5742 1.0473 0.1385 0.2871 0.5237 0.0693 0.1435 0.26185.5 0.2503 0.5158 0.9328 0.1252 0.2579 0.4664 0.0626 0.1289 0.23326.0 0.2283 0.4681 0.8408 0.1142 0.2341 0.4204 0.0571 0.1170 0.2102

4.0 4.0 0.6324 1.1205 1.8827 0.3162 0.5602 0.9414 0.1581 0.2801 0.47074.5 0.5533 0.9700 1.6068 0.2766 0.4850 0.8034 0.1383 0.2425 0.40175.0 0.4917 0.8548 1.4003 0.2458 0.4274 0.7001 0.1229 0.2137 0.35015.5 0.4424 0.7640 1.2401 0.2212 0.3820 0.6201 0.1106 0.1910 0.31006.0 0.4020 0.6900 1.1125 0.2010 0.3452 0.5562 0.1005 0.1726 0.2781

3.7. Mean Deviations

The mean deviation about the mean and the mean deviation about the medianare, respectively, defined by

δ1(µ) = 2µF (µ)− 2µ+ 2

∫ ∞µ

zf(z)dz (23)

δ2(M) = 2MF (M)−M − µ+ 2

∫ ∞M

zf(z)dz (24)

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The Poisson-Lomax Distribution 235

Theorem 5. Let X be a random variable distributed according to the PL distribu-tion. Then the mean deviation about the mean, δ1, and the mean deviation aboutthe median, δ2, are given as follows:

δ1(µ) =2

1− e−λ

{µ(e−λ(1+βµ)−α − 1)− α

β

∞∑n=0

λn+1(−1)n

n!

×(

(1 + βµ)1−α(n+1)

1− α(n+ 1)+

(1 + βµ)−α(n+1)

α(n+ 1)

)} (25)

and

δ2(M) =1

1− e−λ

{M(

2e−λ(1+βM)−α − e−α − 1)

+1

β

∞∑n=0

(−1)nλn+1

(n+ 1)!(1− (n+ 1)α)

− 2α

β

∞∑n=0

λn+1(−1)n

n!

((1 + βM)1−α(n+1)

1− α(n+ 1)

+(1 + βM)−α(n+1)

α(n+ 1)

)}(26)

Proof . The proof follows by plugging the density function of the PLD into equa-tion (23) and working out the integration I, where

I =

∫ ∞µ

xg(x)dx =αβλ

1− e−λ

∫ ∞µ

x(1 + βx)−(α+1)e−λ(1+βx)−αdx

Setting y = 1 +βx, so dy = βdx and using the expansion ex =∑∞n=0 x

n/n!, yields

I =−α

β(1− e−λ)

∞∑n=0

λn+1(−1)n

n!

((1 + βµ)1−α(n+1)

1− α(n+ 1)+

(1 + βµ)−α(n+1)

α(n+ 1)

)Substituting I into relation (23) and manipulating the other terms gives directlythe desired result. Similarly, the measure δ2(M) can be obtained.

4. Estimation

In this section we consider maximum likelihood estimation (MLE) to estimatethe involved parameters. Asymptotic distribution of θ = (α, β, λ) are obtainedusing the elements of the inverse Fisher information matrix.

4.1. Maximum Likelihood Estimation

The idea behind the maximum likelihood parameter estimation is to determinethe parameters that maximize the probability (likelihood) of the sample data. For

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236 Bander Al-Zahrani & Hanaa Sagor

this purpose, let X1, X2, · · · , Xn is be random sample from X ∼ PLD(θ), whereθ = (α, β, λ). Then the likelihood function of the observed sample is given by

L(θ;x) =

n∏i=1

f(xi;θ)

=

n∏i=1

λαβ(1 + βxi)−(α+1) e−λ(1+βxi)

−α

(1− e−λ)

=(λαβ)n

(1− e−λ)n

n∏i=1

(1 + βxi)−(α+1) e−λ

∑ni=1(1+βxi)

−α(27)

The log-likelihood function is given by

`(x;α, β, λ) = n ln(α) + n ln(β) + n ln(λ)− (α+ 1)

n∑i=1

ln(1 + βxi)

−λn∑i=1

(1 + βxi)−α − n ln(1− e−λ) (28)

The MLEs of α, β and λ say α, β and λ, respectively, can be worked out by thesolutions of the system of equations obtained by letting the first partial derivativesof the total log-likelihood equal to zero with respect to α, β and λ. Therefore, thesystem of equations is as follows:

∂`

∂α=n

α−

n∑i=1

ln(1 + βxi) + λ

n∑i=1

(1 + βxi)−α ln(1 + βxi) = 0

∂`

∂β=n

β− (α+ 1)

n∑i=1

xi1 + βxi

+ αλ

n∑i=1

xi(1 + βxi)−(α+1) = 0

∂`

∂λ=n

λ−

n∑i=1

(1 + βxi)−α − n

(eλ − 1)= 0

For simplicity, we define Ai to be as Ai = 1 + βxi. Thus, we have

α = n

[n∑i=1

ln(Ai) (1− λA−αi )

]−1

(29)

β = n

[n∑i=1

xiAi

(α+ 1− αλA−αi )

]−1

(30)

λ = n

[n∑i=1

A−αi +n

eλ − 1

]−1

(31)

The solutions of nonlinear equations (29), (30) and (31) are complicated to obtain,therefore an iterative procedure is applied to solve these equations numerically.

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The Poisson-Lomax Distribution 237

4.2. Asymptotic Distribution

We obtain the asymptotic distribution of θ = (α, β, λ). The asymptotic vari-ances of MLEs are given by the elements of the inverse of the Fisher informationmatrix. The Fisher information matrix of θ, denoted by J(θ) = E(I,θ), whereIij , i, j = 1, 2, 3 is the observed information matrix. The second partial derivativesof the maximum likelihood function are given as the following:

I11 = − n

α2− λ

n∑i=1

(1 + βxi)−α[ln(1 + βxi)]

2

= − n

α2− λ

n∑i=1

A−αi [ln(Ai)]2

I12 = I21 =

n∑i=1

−xi(1 + βxi)

+ λ

n∑i=1

xi(1 + βxi)−(α+1) [1− α ln(1 + βxi)]

=

n∑i=1

[xiAi

(−1− λαA−αi ln(Ai) + λA−αi

)]

I13 = I31 =

n∑i=1

(1 + βxi)−α ln(1 + βxi) =

n∑i=1

A−αi ln(Ai)

I22 = − n

β2+ (α+ 1)

n∑i=1

x2i

(1 + βxi)2− λα(α+ 1)

n∑i=1

x2i (1 + βxi)

−α

(1 + βxi)2

= − n

β2+ (α+ 1)

n∑i=1

(xiAi

)2 (1− λαA−αi

)I23 = I32 = α

n∑i=1

xi(1 + βxi)−(α+1) = α

n∑i=1

xiA−(α+1)i

I33 = − n

λ2+

neλ

(eλ − 1)2

The exact mathematical expressions for J(θ) = E(I,θ) are complicated toobtain. Therefore, the observed Fisher information matrix can be used instead ofthe Fisher information matrix. The variance-covariance matrix may be approxi-mated as V ij = I−1

ij . The asymptotic distribution of the maximum likelihood canbe written as follows (see Miller 1981).[

(α− α), (β − β), (λ− λ)]∼ N3 (0,V ) (32)

Since V involves the parameters α, β and λ, we replace the parameters by thecorresponding MLEs in order to obtain an estimate of V , which is denoted by V .By using (32), approximate 100(1 − ϑ)% confidence intervals for α, β and λ aredetermined, respectively, as

α± Zϑ/2√V 11, β ± Zϑ/2

√V 22, λ± Zϑ/2

√V 33,

where Zϑ is the upper 100ϑ-th percentile of the standard normal distribution.

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238 Bander Al-Zahrani & Hanaa Sagor

In the order to numerically illustrate the estimation of the involved parameters,we have simulated the ML estimators for different sample sizes. The calculationof the estimation is based on 10, 000 simulated samples from the standard PLD.Table 4 shows the MLEs, mean squared errors (MSE) and 95% confidence limits(LCL & UCL ) for the parameters α, β, and λ. The true values of the parametersused for simulation were α = 1, β = 1, and λ = 2. It is observed that when thesample size n increases, the MLE of α and λ decrease to approach the true onewhile the MLEs of the parameters β increase.

Table 4: Simulation study: parameter values used for simulation (TRUE) α = 1, β =1, λ = 2, MLEs, mean squared errors (MSE) and 95% confidence limits (LCL& UCL ) for the parameters.

95% Confi. LimitsParameters n Estimates MSE LCL UCL

α 20 1.10868 0.05159 -2.00901 4.2263730 1.08199 0.03129 -1.12927 3.2932640 1.06866 0.02202 -0.62073 2.7580750 1.06119 0.01762 -1.43696 3.5593560 1.05224 0.01431 0.10648 1.9980070 1.04646 0.01203 0.15111 1.9418180 1.04378 0.01034 0.01529 2.0722790 1.03915 0.00871 0.18454 1.89376100 1.03811 0.00791 0.21745 1.85878200 1.02512 0.00375 0.30619 1.74405

β 20 0.94360 0.05699 0.52240 1.3648030 0.94997 0.03854 0.60608 1.2938740 0.95472 0.03019 0.65637 1.2530850 0.96011 0.02421 0.69225 1.2279760 0.96078 0.02043 0.71629 1.2052770 0.96329 0.01748 0.73662 1.1899780 0.96387 0.01600 0.75180 1.1759490 0.96371 0.01401 0.76389 1.16353100 0.97031 0.01216 0.77951 1.16110200 0.97528 0.00683 0.83990 1.11065

λ 20 2.07641 0.05612 0.38236 3.7704530 2.05300 0.03373 0.67893 3.4270640 2.03975 0.02294 0.85301 3.2264950 2.03150 0.01773 0.97162 3.0913760 2.02744 0.01478 1.06066 2.9942270 2.02349 0.01221 1.12896 2.9180180 2.02025 0.01077 1.18387 2.8566290 2.01885 0.00929 1.23050 2.80719100 2.01723 0.00845 1.26951 2.76495200 2.00944 0.00388 1.48125 2.53762

5. Application

We have considered a dataset corresponding to remission times (in months) ofa random sample of 128 bladder cancer patients given in Lee & Wang (2003). The

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The Poisson-Lomax Distribution 239

data are given as follows: 0.08, 2.09, 3.48, 4.87, 6.94 , 8.66, 13.11, 23.63, 0.20, 2.23,3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50,2.46 , 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76,26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34,14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41,7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33,5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34,5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25,8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73,2.07, 3.36, 6.93, 8.65, 12.63, 22.69. We have fitted the Poisson-Lomax distributionto the dataset using MLE, and compared the proposed PLD with Lomax, extendedLomax and Lomax-Logarithmic distributions.

The model selection is carried out using the AIC (Akaike information criterion),the BIC (Bayesian information criterion), the CAIC (consistent Akaike informationcriteria) and the HQIC (Hannan-Quinn information criterion).

AIC = −2l(θ) + 2q,

BIC = −2l(θ) + q log(n),

HQIC = −2l(θ) + 2q log(log(n)),

CAIC = −2l(θ) + 2qnn−q−1

(33)

where l(θ) denotes the log-likelihood function evaluated at the maximum likelihoodestimates, q is the number of parameters, and n is the sample size. Here we letθ denote the parameters, i.e., θ = (α, β, λ). An iterative procedure is applied tosolve equations (29), (30) and (31) and consequently obtain θ = (α = 2.8737, β =8.2711, p = 3.3515). At these values we calculate the log-likelihood function givenby (28) and apply relation (33). The model with minimum AIC ( or BIC, CAICand HQIC) value is chosen as the best model to fit the data. From Table 5, weconclude that the PLD is best comparable to the Lomax, extended Lomax andLomax-Logarithmic models.

Table 5: MLEs (standard errors in parentheses) and the measures AIC, BIC, HQICand CAIC.

Estimates MeasuresModels α β γ λ AIC BIC HQIC CAICLomax 13.9384 121.0222 831.67 837.37 833.98 831.76

(15.3837) (142.6940)MOEL 23.7437 2.0487 2.2818 825.08 833.64 828.56 825.27

(35.8106) (2.5891) (0.5551)PLD 2.8737 8.2711 3.3515 824.77 833.33 828.25 824.96

(0.8869) (4.8795) (1.0302)

For an ordered random sample, X1, X2, . . . , Xn, from PLD(α, β, λ), where theparameters α, β and λ are unknown, the Kolmogorov-Smirnov Dn, Cramér-vonMises W 2

n , Anderson and Darling A2n, Watson U2

n and Liao-Shimokawa L2n tests

statistics are given as follows: (For details see e.g. Al-Zahrani (2012) and referencestherein).

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240 Bander Al-Zahrani & Hanaa Sagor

Dn = maxi

[i

n−GPL(xi, α, β, λ), GPL(xi, α, β, λ)− i− 1

n

]W 2n =

1

12n+

n∑i=1

[GPL(xi, α, β, λ)− 2i− 1

2n

]2

A2n = −n− 1

n

n∑i=1

(2i− 1)[log(GPL(xi, α, β, λ)) + log(1−GPL(xi, α, β, λ))

]2U2n = W 2

n +

n∑i=1

[GPL(xi, α, β, λ)

n− 1

2

]2

Ln =1√n

n∑i=1

maxi

[in −GPL(xi, α, β, λ), GPL(xi, α, β, λ)− i−1

n

]√GPL(xi, α, β, λ)[1−GPL(xi, α, β, λ)]

.

Table 6 indicates that the test statisticsDn,W 2n , A2

n, U2n and Ln have the small-

est values for the data set under PLD model with regard to the other models. Theproposed model offers an attractive alternative to the Lomax, Lomax-Logarithmicand extended Lomax models. Figure 3 displays the empirical and fitted densitiesfor the data. Estimated survivals for data are shown in Figure 4. The Poisson-Lomax distribution approximately provides an adequate fit for the data. Thequantile-quantile or Q-Q plot is used to check the validity of the distributionalassumption for the data. Figure 5 shows that the data seems to follow a PLDreasonably well, except some points on extreme.

Table 6: Goodness-of-fit tests.Statistics

Distribution Dn W 2n A2

n U2n Ln

Lomax 0.0967 0.2126 1.3768 31.7017 1.0594MOEL 0.0302 0.0151 0.0926 31.5177 0.3728LLD 0.0821 0.1274 0.8739 31.6200 0.8491PLD 0.0281 0.0134 0.0835 31.5164 0.3567

6. Concluding Remarks

In this paper we have proposed a new distribution, referred to as the PLD. Amathematical treatment of the proposed distribution including explicit formulasfor the density and hazard functions, moments, order statistics, and mean and me-dian deviations have been provided. The estimation of the parameters has beenapproached by maximum likelihood. Also, the asymptotic variance-covariance ma-trix of the estimates has been obtained. Finally, a real data set was analyzed toshow the potential of the proposed PLD. The result indicates that the PLD maybe used for a wider range of statistical applications. Further study can be con-ducted on the proposed distribution. Here, we mention some of possible directions

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The Poisson-Lomax Distribution 241

which are still open for further works. The problem of parameter estimation canbe studied using e.g. Bayesian approach and making future prediction. The pa-rameters of the proposed distribution can be estimated based on censored data.Some recurrence relations can be established for the single moments and productmoments of order statistics.

Remission Times

Den

sity

Fun

ctio

n

0 20 40 60 80

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Empirical DensityLomaxExtended LomaxPoisson−Lomax

Figure 3: Estimated densities for bladder cancer data.

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Remission Times

Est

imat

ed S

urvi

val F

unct

ion

EmpricalLomaxExtended LomaxPoisson−Lomax

Figure 4: Estimated survivals for bladder cancer data.

Acknowledgments

The authors are grateful to the Editor and the anonymous referees for theirvaluable comments and suggestions that improved the presentation of the paper.This study is a part of the Master Thesis of the second named author whose workwas supervised by the first named author.

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242 Bander Al-Zahrani & Hanaa Sagor

0.0 0.4 0.8 1.20

1020

3040

PLD Q−Q Plot

Theoretical quantile

Sam

ple

quan

tile

0 5 10 15 20

010

2030

40

MOEL Q−Q Plot

Theoretical quantile

Sam

ple

quan

tile

0.000 0.002 0.004

010

2030

40

Lomax Q−Q Plot

Theoretical quantile

Sam

ple

quan

tile

−2 −1 0 1 20

1020

3040

Normal Q−Q Plot

Theoretical Quantiles

Sam

ple

Qua

ntile

s

Figure 5: The Q-Q plot for bladder cancer data.

[Recibido: noviembre de 2013 — Aceptado: abril de 2014

]

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